XMM-Newton spectra of five red, 2MASS AGN, selected from a sample observed by Chandra to be relatively X-ray bright and to cover a range of hardness ratios, confirm the presence of substantial absorbing material in three sources with optical classifications ranging from Type 1 to Type 2. A flat (hard), power law continuum is observed in the other two. The combination of X-ray absorption and broad optical emission lines suggests either a small (nuclear) absorber or a favored viewing angle so as to cover the X-ray source but not the broad emission line region (BELR). A soft excess is detected in all three Type 1 sources. We speculate that this may arise in an extended region of ionised gas, perhaps linked with the polarised (scattered) optical light present in these sources. The spectral complexity revealed by XMM-Newton emphasizes the limitations of the low S/N chandra data. The new results strengthen our earlier conclusions that the observed X-ray continua of red AGN are unusually hard at energies >2 keV. Their observed spectra are consistent with contributing significantly to the missing hard/absorbed population of the Cosmic X-ray Background (CXRB) although their intrinsic power law slopes are typical of broad-line (Type 1) AGN (Gamma ~1.7-1.9). This suggests that the missing X-ray-absorbed CXRB population may include Type 1 AGN/QSOs in addition to the Type 2 AGN generally assumed.